Topic 11 Sorting and Searching

1. Topic 11 Sorting and Searching "There's nothing in your head the sorting hat can't see. So try me on and I will tell you where you ought to be." -The Sorting Hat, Harry Potter and the Sorcerer's Stone Sorting and Searching

2. Sorting and Searching • Fundamental problems in computer science and programming • Sorting done to make searching easier • Multiple different algorithms to solve the same problem • How do we know which algorithm is "better"? • Look at searching first • Examples will use arrays of ints to illustrate algorithms Sorting and Searching

3. Searching Sorting and Searching

4. Searching • Given a list of data find the location of a particular value or report that value is not present • linear search • intuitive approach • start at first item • is it the one I am looking for? • if not go to next item • repeat until found or all items checked • If items not sorted or unsortable this approach is necessary Sorting and Searching

5. Linear Search /* pre: list != null post: return the index of the first occurrence of target in list or -1 if target not present in list */ public int linearSearch(int[] list, int target) { for(int i = 0; i < list.length; i++) if( list[i] == target ) return i; return -1; } Sorting and Searching

6. Linear Search, Generic /* pre: list != null, target != null post: return the index of the first occurrence of target in list or -1 if target not present in list */ public int linearSearch(Object[] list, Object target) { for(int i = 0; i < list.length; i++) if( list[i] != null && list[i].equals(target) ) return i; return -1; } T(N)? Big O? Best case, worst case, average case? Sorting and Searching

7. Attendance Question 1 • What is the average case Big O of linear search in an array with N items, if an item is present? • O(N) • O(N2) • O(1) • O(logN) • O(NlogN) Sorting and Searching

8. Searching in a Sorted List • If items are sorted then we can divide and conquer • dividing your work in half with each step • generally a good thing • The Binary Search on List in Ascending order • Start at middle of list • is that the item? • If not is it less than or greater than the item? • less than, move to second half of list • greater than, move to first half of list • repeat until found or sub list size = 0 Sorting and Searching

9. Binary Search list low item middle item high item Is middle item what we are looking for? If not is it more or less than the target item? (Assume lower) list low middle high item item item and so forth… Sorting and Searching

10. Binary Search in Action 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 public static int bsearch(int[] list, int target){ int result = -1; int low = 0; int high = list.length - 1; int mid; while( result == -1 && low <= high ) { mid = low + ((high - low) / 2); if( list[mid] == target ) result = mid; else if( list[mid] < target) low = mid + 1; else high = mid - 1; } return result; } // mid = ( low + high ) / 2; // may overflow!!!// or mid = (low + high) >>> 1; using bitwise op Sorting and Searching

11. Trace When Key == 3Trace When Key == 30Variables of Interest? Sorting and Searching

12. Attendance Question 2 What is the worst case Big O of binary search in an array with N items, if an item is present? O(N) O(N2) O(1) O(logN) O(NlogN) Sorting and Searching

13. Generic Binary Search public static int bsearch(Comparable[] list, Comparable target){ int result = -1; int low = 0; int high = list.length - 1; int mid; while( result == -1 && low <= high ) { mid = low + ((high - low) / 2); if( target.equals(list[mid]) ) result = mid; else if(target.compareTo(list[mid]) > 0) low = mid + 1; else high = mid - 1; } return result; } Sorting and Searching

14. Recursive Binary Search public static int bsearch(int[] list, int target){ return bsearch(list, target, 0, list.length – 1); } public static int bsearch(int[] list, int target, int first, int last){ if( first <= last ){ int mid = low + ((high - low) / 2); if( list[mid] == target ) return mid; else if( list[mid] > target ) return bsearch(list, target, first, mid – 1); else return bsearch(list, target, mid + 1, last); } return -1; } Sorting and Searching

15. Other Searching Algorithms • Interpolation Search • more like what people really do • Indexed Searching • Binary Search Trees • Hash Table Searching • Grover's Algorithm (Waiting for quantum computers to be built) • best-first • A* Sorting and Searching

16. Sorting Sorting and Searching

17. Sorting FunWhy Not Bubble Sort? Sorting and Searching

18. Sorting • A fundamental application for computers • Done to make finding data (searching) faster • Many different algorithms for sorting • One of the difficulties with sorting is working with a fixed size storage container (array) • if resize, that is expensive (slow) • The "simple" sorts run in quadratic time O(N2) • bubble sort • selection sort • insertion sort Sorting and Searching

19. Stable Sorting • A property of sorts • If a sort guarantees the relative order of equal items stays the same then it is a stable sort • [71, 6, 72, 5, 1, 2, 73, -5] • subscripts added for clarity • [-5, 1, 2, 5, 6, 71, 72, 73] • result of stable sort • Real world example: • sort a table in Wikipedia by one criteria, then another • sort by country, then by major wins Sorting and Searching

20. Selection sort • Algorithm • Search through the list and find the smallest element • swap the smallest element with the first element • repeat starting at second element and find the second smallest element public static void selectionSort(int[] list) { int min; int temp; for(int i = 0; i < list.length - 1; i++) { min = i; for(int j = i + 1; j < list.length; j++) if( list[j] < list[min] ) min = j; temp = list[i]; list[i] = list[min]; list[min] = temp; } } Sorting and Searching

21. Selection Sort in Practice 44 68 191 119 119 37 83 82 191 45 158 130 76 153 39 25 What is the T(N), actual number of statements executed, of the selection sort code, given a list of N elements? What is the Big O? Sorting and Searching

22. Generic Selection Sort • public void selectionSort(Comparable[] list){ int min; Comparable temp; for(int i = 0; i < list.length - 1; i++) { { min = i; for(int j = i + 1; j < list.length; j++) • if( list[min].compareTo(list[j]) > 0 ) min = j; temp = list[i]; list[i] = list[min]; list[min] = temp; }} • Best case, worst case, average case Big O? Sorting and Searching

23. Attendance Question 3 Is selection sort always stable? Yes No Sorting and Searching

24. Insertion Sort • Another of the O(N^2) sorts • The first item is sorted • Compare the second item to the first • if smaller swap • Third item, compare to item next to it • need to swap • after swap compare again • And so forth… Sorting and Searching

25. Insertion Sort Code • public void insertionSort(int[] list) • { int temp, j; • for(int i = 1; i < list.length; i++) • { temp = list[i]; • j = i; • while( j > 0 && temp < list[j - 1]) • { // swap elements • list[j] = list[j - 1]; • list[j - 1] = temp; • j--; • } • } • } • Best case, worst case, average case Big O? Sorting and Searching

26. Attendance Question 4 • Is the version of insertion sort shown always stable? • Yes • No Sorting and Searching

27. Comparing Algorithms • Which algorithm do you think will be faster given random data, selection sort or insertion sort? • Why? Sorting and Searching

28. Sub Quadratic Sorting Algorithms Sub Quadratic means having a Big O better than O(N2) Sorting and Searching

29. ShellSort • Created by Donald Shell in 1959 • Wanted to stop moving data small distances (in the case of insertion sort and bubble sort) and stop making swaps that are not helpful (in the case of selection sort) • Start with sub arrays created by looking at data that is far apart and then reduce the gap size Sorting and Searching

30. ShellSort in practice 46 2 83 41 102 5 17 31 64 49 18 Gap of five. Sort sub array with 46, 5, and 185 2 83 41 102 18 17 31 64 49 46 Gap still five. Sort sub array with 2 and 175 2 83 41 102 18 17 31 64 49 46 Gap still five. Sort sub array with 83 and 315 2 31 41 102 18 17 83 64 49 46 Gap still five Sort sub array with 41 and 645 2 31 41 102 18 17 83 64 49 46 Gap still five. Sort sub array with 102 and 495 2 31 41 49 18 17 83 64 102 46 Continued on next slide: Sorting and Searching

31. Completed Shellsort 5 2 31 41 49 18 17 83 64 102 46 Gap now 2: Sort sub array with 5 31 49 17 64 465 2 17 41 31 18 46 83 49 102 64 Gap still 2: Sort sub array with 2 41 18 83 1025 2 17 18 31 41 46 83 49 102 64 Gap of 1 (Insertion sort) 2 5 17 18 31 41 46 49 64 83 102 Array sorted Sorting and Searching

32. Shellsort on Another Data Set 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 44 68 191 119 119 37 83 82 191 45 158 130 76 153 39 25 Initial gap = length / 2 = 16 / 2 = 8 initial sub arrays indices: {0, 8}, {1, 9}, {2, 10}, {3, 11}, {4, 12}, {5, 13}, {6, 14}, {7, 15}next gap = 8 / 2 = 4 {0, 4, 8, 12}, {1, 5, 9, 13}, {2, 6, 10, 14}, {3, 7, 11, 15}next gap = 4 / 2 = 2 {0, 2, 4, 6, 8, 10, 12, 14}, {1, 3, 5, 7, 9, 11, 13, 15} final gap = 2 / 2 = 1 Sorting and Searching

33. ShellSort Code public static void shellsort(Comparable[] list){ Comparable temp; boolean swap; for(int gap = list.length / 2; gap > 0; gap /= 2) for(int i = gap; i < list.length; i++) { Comparable tmp = list[i]; int j = i; for( ; j >= gap && tmp.compareTo( list[j - gap] ) < 0; j -= gap ) list[ j ] = list[ j - gap ]; list[ j ] = tmp; }} Sorting and Searching

34. Comparison of Various Sorts times in milliseconds Sorting and Searching

35. Quicksort • Invented by C.A.R. (Tony) Hoare • A divide and conquer approach that uses recursion • If the list has 0 or 1 elements it is sorted • otherwise, pick any element p in the list. This is called the pivot value • Partition the list minus the pivot into two sub lists according to values less than or greater than the pivot. (equal values go to either) • return the quicksort of the first list followed by the quicksort of the second list Sorting and Searching

36. Quicksort in Action 39 23 17 90 33 72 46 79 11 52 64 5 71 Pick middle element as pivot: 46 Partition list 23 17 5 33 39 11 46 79 72 52 64 90 71 quick sort the less than list Pick middle element as pivot: 33 23 17 5 11 33 39 quicksort the less than list, pivot now 5 {} 5 23 17 11 quicksort the less than list, base case quicksort the greater than list Pick middle element as pivot: 17and so on…. Sorting and Searching

37. Quicksort on Another Data Set 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 44 68 191 119 119 37 83 82 191 45 158 130 76 153 39 25 Big O of Quicksort? Sorting and Searching

38. public static void swapReferences( Object[] a, int index1, int index2 ) { Object tmp = a[index1]; a[index1] = a[index2]; a[index2] = tmp; } public void quicksort( Comparable[] list, int start, int stop ) { if(start >= stop) return; //base case list of 0 or 1 elements int pivotIndex = (start + stop) / 2; // Place pivot at start position swapReferences(list, pivotIndex, start); Comparable pivot = list[start]; // Begin partitioning int i, j = start; // from first to j are elements less than or equal to pivot // from j to i are elements greater than pivot // elements beyond i have not been checked yet for(i = start + 1; i <= stop; i++ ) { //is current element less than or equal to pivot if(list[i].compareTo(pivot) <= 0) { // if so move it to the less than or equal portion j++; swapReferences(list, i, j); } } //restore pivot to correct spot swapReferences(list, start, j); quicksort( list, start, j - 1 ); // Sort small elements quicksort( list, j + 1, stop ); // Sort large elements } Sorting and Searching

39. Attendance Question 5 • What is the best case and worst case Big O of quicksort?Best Worst • O(NlogN) O(N2) • O(N2) O(N2) • O(N2) O(N!) • O(NlogN) O(NlogN) • O(N) O(NlogN) Sorting and Searching

40. Quicksort Caveats • Average case Big O? • Worst case Big O? • Coding the partition step is usually the hardest part Sorting and Searching

41. Attendance Question 6 • You have 1,000,000 items that you will be searching. How many searches need to be performed before the data is changed to make sorting worthwhile? • 10 • 40 • 1,000 • 10,000 • 500,000 Sorting and Searching

42. Merge Sort Algorithm • If a list has 1 element or 0 elements it is sorted • If a list has more than 2 split into into 2 separate lists • Perform this algorithm on each of those smaller lists • Take the 2 sorted lists and merge them together Don Knuth cites John von Neumann as the creatorof this algorithm Sorting and Searching

43. Merge Sort When implementing one temporary array is used instead of multiple temporary arrays. Why? Sorting and Searching

44. Merge Sort code /** * perform a merge sort on the data in c * @param c c != null, all elements of c * are the same data type */ public static void mergeSort(Comparable[] c) { Comparable[] temp = new Comparable[ c.length ]; sort(c, temp, 0, c.length - 1); } private static void sort(Comparable[] list, Comparable[] temp, int low, int high) { if( low < high){ int center = (low + high) / 2; sort(list, temp, low, center); sort(list, temp, center + 1, high); merge(list, temp, low, center + 1, high); } } Sorting and Searching

45. Merge Sort Code private static void merge( Comparable[] list, Comparable[] temp, int leftPos, int rightPos, int rightEnd){ int leftEnd = rightPos - 1; int tempPos = leftPos; int numElements = rightEnd - leftPos + 1; //main loop while( leftPos <= leftEnd && rightPos <= rightEnd){ if( list[ leftPos ].compareTo(list[rightPos]) <= 0){ temp[ tempPos ] = list[ leftPos ]; leftPos++; } else{ temp[ tempPos ] = list[ rightPos ]; rightPos++; } tempPos++; } //copy rest of left half while( leftPos <= leftEnd){ temp[ tempPos ] = list[ leftPos ]; tempPos++; leftPos++; } //copy rest of right half while( rightPos <= rightEnd){ temp[ tempPos ] = list[ rightPos ]; tempPos++; rightPos++; } //Copy temp back into list for(int i = 0; i < numElements; i++, rightEnd--) list[ rightEnd ] = temp[ rightEnd ]; } Sorting and Searching

46. Final Comments • Language libraries often have sorting algorithms in them • Java Arrays and Collections classes • C++ Standard Template Library • Python sort and sorted functions • Hybrid sorts • when size of unsorted list or portion of array is small use insertion sort, otherwise use O(N log N) sort like Quicksort of Mergesort • Many other sorting algorithms exist. Sorting and Searching